I am not sure if this is a card-specific thing, driver-specific or if I am doing something wrong.

Here is the problem:
The card (ATI Radeon HD 4300/4500 Series) does 4x multisampling. The quality of the output is lower when drawing to the screen
http://chattypics.com/files/screen_q1oju76nc2.png
than it is if you draw to a multisampled FBO
http://chattypics.com/files/fbo_e4amht2pnv.png

The problem doesn't appear on my other machine which has an NVidia card.
But then, that card does 16x multisampling and maybe the difference there. But just isn't noticeable.